Group #22560253 — subgroups

Subgroups — #22560253

Order 8576.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14524050	C₂	1	1	yes	maximal, characteristic
2	#16863432	C₂	1	2	yes	maximal, complemented
3	#16863370	C₂	1	1	yes	maximal, characteristic
4	#16863152	C₂	1	2	yes	maximal, complemented
5	#410×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524050	—	2	2	no	—
7	#14524050	—	2	2	no	—
8	C₂×#11761612	C₄	1	2	yes	complemented
9	C₂×#11761612	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	#14524002	C₂²	1	2	yes	—
13	#14523976	—	2	2	no	—
14	#14523974	—	2	2	no	—
15	#14523958	—	2	2	no	—
16	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#117×C₆₇	C₂²	1	2	yes	—
18	#99×C₆₇	—	2	2	no	—
19	#11761612	—	2	2	no	—
20	#11761612	—	2	4	no	—
21	C₂×C₆₇⋊C₈	—	2	2	no	—
22	C₂×C₆₇⋊C₈	—	2	2	no	—
23	C₂×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
27	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
28	C₄⋊Dic₆₇	—	2	2	no	—
29	C₂²⋊Dic₆₇	—	2	2	no	—
30	C₄×Dic₆₇	—	2	2	no	—
31	#41×C₆₇	—	2	2	no	—
32	#41×C₆₇	—	2	2	no	—
33	C₂×Q₈×C₆₇	Q₈	1	1	yes	characteristic
34	C₂×D₄×C₆₇	D₄	1	2	yes	—
35	C₂×D₄×C₆₇	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂²⋊C₄×C₆₇	—	2	2	no	—
40	C₄²×C₆₇	—	2	2	no	—
41	C₂³×C₆₇	C₄⋊C₄	1	2	yes	—
42	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₆₇	—	2	2	no	—
44	D₄×C₆₇	—	2	4	no	—
45	D₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	2	2	no	—
47	C₂×C₄×C₆₇	—	2	2	no	—
48	C₂×C₄×C₆₇	—	4	2	no	—
49	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	#41	1	2	yes	—
54	C₂×Dic₆₇	—	4	2	no	—
55	C₆₇⋊C₈	—	2	2	no	—
56	C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₆₇	—	2	2	no	—
60	C₂²×C₆₇	—	4	1	no	—
61	C₂²×C₆₇	—	4	2	no	—
62	C₂²×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₆₇	#121	1	2	yes	—
64	C₄×C₆₇	—	2	2	no	—
65	C₄×C₆₇	—	8	2	no	—
66	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₆₇	#117	1	1	yes	characteristic
68	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
69	Dic₆₇	—	8	2	no	—
70	C₂×C₆₇	—	2	2	no	—
71	C₂×C₆₇	—	2	1	no	—
72	C₂×C₆₇	—	4	2	no	—
73	C₂×C₆₇	#380	1	1	yes	characteristic
74	#1812	—	67	1	no	maximal, Sylow
75	C₆₇	#1812	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	—	67	1	no	—
77	#427	—	67	2	no	—
78	#410	—	67	1	no	—
79	#410	D₆₇	1	1	yes	characteristic
80	H(3,ℤ/4)	—	67	2	no	—
81	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
82	#131	—	134	2	no	—
83	#131	—	134	2	no	—
84	C₂×M₁₆	—	67	2	no	—
85	C₂×M₁₆	—	67	1	no	—
86	C₂²×C₈	—	67	1	no	—
87	C₂²×C₈	—	67	2	no	—
88	#117	—	67	2	no	—
89	#117	C₂×D₆₇	1	2	yes	—
90	#107	—	134	2	no	—
91	#106	—	134	2	no	—
92	#99	—	2	2	no	—
93	#99	—	134	2	no	—
94	#41	—	2	2	no	—
95	#41	—	2	2	no	—
96	C₂×Q₈	Dic₁₃₄	1	1	yes	characteristic
97	C₂×D₄	Dic₁₃₄	1	1	yes	characteristic
98	C₂×D₄	C₂²⋊D₆₇	1	2	yes	—
99	C₂²×C₄	C₄×D₆₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
101	M₁₆	—	134	4	no	—
102	M₁₆	—	134	2	no	—
103	C₂×C₈	—	67	1	no	—
104	C₂×C₈	—	67	1	no	—
105	C₂×C₈	—	67	2	no	—
106	C₂×C₈	—	67	2	no	—
107	C₂×C₈	—	67	2	no	—
108	C₂×C₈	—	134	2	no	—
109	C₂×C₈	—	134	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	134	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₂²⋊C₄	—	134	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	134	2	no	—
116	C₂³	Dic₆₇⋊C₄	1	2	yes	—
117	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
118	Q₈	—	2	2	no	—
119	D₄	—	2	4	no	—
120	D₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	268	2	no	—
125	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₆₇	1	2	yes	—
128	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
129	C₂×C₄	#11761627	1	2	yes	—
130	C₈	—	67	1	no	—
131	C₈	—	67	1	no	—
132	C₈	—	67	2	no	—
133	C₈	—	134	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
138	C₂²	#14524015	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	2	no	—
141	C₄	—	536	2	no	—
142	C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
143	C₄	#14523999	1	1	yes	characteristic
144	C₄	#14524020	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	#16863254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524050

C₂

yes

maximal, characteristic

#16863432

C₂

yes

maximal, complemented

#16863370

C₂

yes

maximal, characteristic

#16863152

C₂

yes

maximal, complemented

#410×C₆₇

C₂

yes

maximal, characteristic

#14524050

—

#14524050

—

C₂×#11761612

C₄

yes

complemented

C₂×#11761612

C₂²

yes

characteristic

C₂²×C₆₇⋊C₈

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524002

C₂²

yes

—

#14523976

—

#14523974

—

#14523958

—

C₂×#41×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

#99×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

Q₈

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₄⋊C₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₆₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₆₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₇

#121

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

#117

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic

H(3,ℤ/4)

—

C₂×#41

C₂×D₆₇

yes

characteristic

#131

—

134

—

#131

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₆₇

yes

—

#107

—

134

—

#106

—

134

—

#99

—

#99

—

134

—

#41

—

#41

—

C₂×Q₈

Dic₁₃₄

yes

characteristic

C₂×D₄

Dic₁₃₄

yes

characteristic

C₂×D₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₄×D₆₇

yes

—

100

C₂²×C₄

C₂²×D₆₇

yes

characteristic

101

M₁₆

—

134

—

102

M₁₆

—

134

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

134

—

109

C₂×C₈

—

134

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

134

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

134

—

114

C₄²

—

115

C₄²

—

134

—

116

C₂³

Dic₆₇⋊C₄

yes

—

117

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

268

—

125

C₂×C₄

Dic₆₇⋊C₄

yes

—

126

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

127

C₂×C₄

D₄×D₆₇

yes

—

128

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

129

C₂×C₄

#11761627

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×Dic₆₇⋊C₄

yes

characteristic

138

C₂²

#14524015

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

536

—

142

C₄

C₂²⋊C₄×D₆₇

yes

characteristic

143

C₄

#14523999

yes

characteristic

144

C₄

#14524020

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#16863254

yes

characteristic

	subgroup	G/H
center	C₄	#14523999
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#410×C₆₇	C₂
socle	C₂×C₆₇	#380
hypercenter	#410	D₆₇
hypocenter	C₆₇	#1812
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#14523999

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#410×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

N	H ≅ G/N	type
2. #16863432	C₂	semidirect
4. #16863152	C₂	semidirect
8. C₂×#11761612	C₄	semidirect
10. C₂²×C₆₇⋊C₈	C₄	semidirect
75. C₆₇	#1812	semidirect

H ≅ G/N

type

2. #16863432

C₂

semidirect

4. #16863152

C₂

semidirect

8. C₂×#11761612

C₄

semidirect

10. C₂²×C₆₇⋊C₈

C₄

semidirect

75. C₆₇

#1812

semidirect

Subgroups — #22560253

Conjugacy classes of subgroups

Complements